Q21GATE 2013MCQ

All members in the rigid-jointed frame shown are prismatic and have the same flexural stiffness EI. Find the magnitude of the bending moment at Q (in kNm) due to the given loading. __________

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📖 Explanation

$\therefore W_{Q} = 25\text{kNm}$

Q23GATE 2006MCQ

Carry-over factor $C_{AB}$ for the beam shown in the figure below is

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📖 Explanation

Applying a moment M at A . Let reaction at A and B is $V_{A}$ and $V_{B}$ respectively, and carryover moment be $M_{B}.$

\begin{aligned} \text{Taking }M_{c}&=0 \quad \text{(from left) }\\ V_{A} \times L-M &=0 \\ \Rightarrow_{A} &=\frac{M}{L} \\ \therefore V_{B}&=-\frac{M}{L} \\ \text{Now, taking }M_{C}&=0 \text{ from right,}\\ M_{B}-V_{B} \times L &=0 \\ \Rightarrow \quad M_{B} &=\left(\frac{-M}{L}\right) \times L \\ M_{B} &=-M \\ \text { Carryover factor } &=\frac{\mid \text { Carryover moment } \mid}{\mid \text { Applied moment } \mid} \\ \Rightarrow &=\frac{M}{M}=1 \end{aligned}

Q24GATE 2005MCQ

Match the following:

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Q25GATE 2005MCQ

All members of the frame shown below have the same flexural rigidity EI and length L. If a moment M is applied at joint B, the rotation of the joint is

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📖 Explanation

\begin{aligned} \text{For }A B, \quad \theta_{B}&=\frac{M L}{4 E I} \\ \Rightarrow \text{ Stiffness at }\quad B&=\frac{4 E I}{L} \\ \text{For }B C \theta_{B}&=\frac{M L}{3 E I} \\ \Rightarrow \text{Stiffness at }\quad B&=\frac{3 E I}{L}\\ \text{For }BD, \quad \theta_{B}&=\frac{M L}{4 E I} \\ \Rightarrow \text{ Stiffness at }\quad B&=\frac{4 E I}{L} \\ \text{Total stiffness of }&\text{the joint}\\ =\frac{4 E I}{L}+\frac{3 E I}{L}+\frac{4 E I}{L}&=\frac{11 E I}{L} \\ \text{Rotation of joint }B&=\frac{M L}{11 E I} \end{aligned}

Q26GATE 2004MCQ

For a linear elastic structural system, minimization of potential energy yields

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📖 Explanation

In a linear elastic structural system for potential energy (U) to be minimum, $\frac{\partial U}{\partial P}=0$ The above equation represents the compatibility condition at the point where load P is acting.

Q27GATE 2004MCQ

The plane frame below is analyzed by neglecting axial deformations. Following statements are made with respect to the analysis : I. Column AB carries axial force only II. Vertical Deflection at the center of beam BC is 1 mm With reference to the above statements, which of the following applies?

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📖 Explanation

Let reaction at A and B be $R_{A}$ and $R_{B}$ respectively. Taking moments about D , we get,

\begin{aligned} R_{A} \times 5-10 \times 5 &\times 2.5=0 \\ \Rightarrow \quad R_{A}&=25 \mathrm{kN}\\ \therefore \quad R_{B}&=(10 \times 5)-25=25 \mathrm{kN} \end{aligned}

Since reaction $R_{A}$ and $R_{D}$ are equal and no horizontal reaction so beam BC will behave as a simply supported beam. Deflected shape is as shown below. Thus deflection at the centre of beam

\begin{array}{l} =\frac{5}{384} \frac{w^{4}}{E I} \\ =\frac{5}{384} \times \frac{10 \times (5)^{4}}{81380} \times 10^{3} \\ =1.00 \mathrm{mm} \end{array}

Column AB will carry only axial force equal to $R_{A}$ i.e. 25kN

Q28GATE 2004MCQ

The unit load method used in structural analysis is

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📖 Explanation

Unit load method is used to find deflection at any point of structure. It is derived from Castigliano's theorem. $\Delta=\frac{\partial U}{\partial Q}=\int M \frac{\partial M}{\partial Q} \frac{d x}{E I}=\int \frac{Mmdx}{E I}$

Q29GATE 2001MCQ

The end moment (in kNm units) developed in the roof level beams in the laterally loaded frame shown below (with all columns having identical cross-sections) according to the Cantilever Method of simplified analysis, is

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📖 Explanation

According to cantilever method, point of contraflexure lies at mid point of each span and direct stresses (axial) in column due to horizontal forces are directly proportional to their distance from the centroidal vertical axis of frame. $\overline{h}$ = CG of frame will be at mid column due to symmetry. $\overline{h}=\frac{1\times0+2\times6+1\times12}{3}=6m$ Since, areas of each column are equal so axial force in each column will be in proportional to their distance from CG . $\frac{V_{1}}{x_{1}}=\frac{V_{3}}{x_{3}}$ As distance of $V_{2}$ from C G=0 $\therefore \quad V_{2}=0$

\begin{aligned} \frac{V_{1}}{6}&=\frac{V_{3}}{6}\\ \Rightarrow \quad V_{1}&=V_{3} \end{aligned}

Take moment about L

\begin{aligned} 30 \times 2 &=V_{1} \times 12 \\ V_{1} &=5 \mathrm{kN} \end{aligned}

and from equilibrium of forces end moment at roof level beam $=7.5 \times 2=15 \mathrm{kNm}$

Q30GATE 2001MCQ

Identify the FALSE statement from the following, pertaining to the methods of structural analysis

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Q31GATE 2001MCQ

The frame below shows three beam elements OA, OB and OC, with identical length L and flexural rigidity El , subject to an external moment M applied at the rigid joint O. The correct set of bending moments $\{M_{OA},M_{OB},M_{OC}\}$ , that develop at O in the three beam elements OA, OB and OC respectively, is

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📖 Explanation

\begin{aligned} M_{O A} &=M \times D F \text { for } O A \\ &=M \times \frac{3}{7}=\frac{3}{7} M\\ M_{O B} &=0 \\ M_{O C} &=M \times D F \text { for } O C \\ &=M \times \frac{4}{7}=\frac{4}{7} M \end{aligned}

Q32GATE 2001MCQ

Identify the correct deflection diagram corresponding to the loading in the plane frame shown below:

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